Hellinger integrals of diffusion processes.

*(English)*Zbl 0598.60042If \(\{X_ n,{\mathcal F}_ n,n\geq 1\}\) is a sequence of measurable spaces and \(P_ n\), \(Q_ n\) are probability measures on \({\mathcal F}_ n\), for each \(n\geq 1\), then the \(\{P_ n,n\geq 1\}\) is said to be contiguous to \(\{Q_ n,n\geq 1\}\) if for any sequence \(\{A_ n\in {\mathcal F}_ n,n\geq 1\}\) satisfying \(P_ n(A_ n)\to 0\), one has \(Q_ n(A_ n)\to 0\). This reduces to the absolutely continuous concept if all the measurable spaces and the measures are independent of n. The natural tools in this measure comparison are the Hellinger integral and the variation norm. The latter is just the variation of the signed measure \((Q_ n-P_ n)\) on \({\mathcal F}_ n\), and the former is defined as:
\[
H_ s(Q_ n-P_ n)=\int_{X_ n}f^ s_ ng_ n^{1-s}d\mu_ n,\quad 0\leq s\leq 1,
\]
where \(\mu_ n=(Q_ n+P_ n)\) and \(f_ n,g_ n\) are the Radon-Nikodým densities of \(P_ n\) and \(Q_ n\) relative to \(\mu_ n\). It can be shown that \(H_ s\) does not depend on the particular dominating measure \(\mu_ n\), \(0\leq H_ s(P_ n,Q_ n)\leq 1\) and \(H_ s\) vanishes iff \(Q_ n\perp P_ n.\)

The effectiveness of the Hellinger integral in probability was shown by S. Kakutani [Ann. Math., II. Ser. 49, 214-224 (1948; Zbl 0030.02303)] and is gainfully employed in the subject ever since. [For more detail and related theory see the reviewer’s book: Foundations of stochastic analysis. (1981; Zbl 0539.60006), Sec. 4.4].

With these concepts the author’s results can be described from the introduction as follows: ”In the present paper Hellinger integrals of distribution laws \(P_{\xi_ i}\) of diffusion processes (which are solutions of the stochastic differential equations) \[ d\xi_ i=a_ i(t,\xi_ i)dt+dW(t), \] i\(=1,2\) are investigated. Here W(t), \(t\geq 0\) is the Wiener process. Our main result is a two-sided estimate of \(H_ s(P_{\xi_ 1},P_{\xi_ 2})\) in terms of the (drift) coefficients \(a_ i\). These estimates are applied to obtain necessary and sufficient conditions for the convergence in variation distance (or norm) and for contiguity. Furthermore, bounds for the error probabilities in the statistical testing problem following a method due to H. Chernoff [Ann. Math. Stat. 23, 493-507 (1952; Zbl 0048.118)] are obtained. All results and proofs remain true in a slightly modified formulation if the processes \(\xi_ i\) have a (common nonconstant) diffusion coefficient, (and different drifts \(a_ i\) as before).”

The effectiveness of the Hellinger integral in probability was shown by S. Kakutani [Ann. Math., II. Ser. 49, 214-224 (1948; Zbl 0030.02303)] and is gainfully employed in the subject ever since. [For more detail and related theory see the reviewer’s book: Foundations of stochastic analysis. (1981; Zbl 0539.60006), Sec. 4.4].

With these concepts the author’s results can be described from the introduction as follows: ”In the present paper Hellinger integrals of distribution laws \(P_{\xi_ i}\) of diffusion processes (which are solutions of the stochastic differential equations) \[ d\xi_ i=a_ i(t,\xi_ i)dt+dW(t), \] i\(=1,2\) are investigated. Here W(t), \(t\geq 0\) is the Wiener process. Our main result is a two-sided estimate of \(H_ s(P_{\xi_ 1},P_{\xi_ 2})\) in terms of the (drift) coefficients \(a_ i\). These estimates are applied to obtain necessary and sufficient conditions for the convergence in variation distance (or norm) and for contiguity. Furthermore, bounds for the error probabilities in the statistical testing problem following a method due to H. Chernoff [Ann. Math. Stat. 23, 493-507 (1952; Zbl 0048.118)] are obtained. All results and proofs remain true in a slightly modified formulation if the processes \(\xi_ i\) have a (common nonconstant) diffusion coefficient, (and different drifts \(a_ i\) as before).”

Reviewer: M.M.Rao

##### MSC:

60G30 | Continuity and singularity of induced measures |

60J60 | Diffusion processes |

60F05 | Central limit and other weak theorems |

62M07 | Non-Markovian processes: hypothesis testing |

62M99 | Inference from stochastic processes |

##### Keywords:

continuity of induced measures; contiguity; hypothesis testing of diffusion processes; error estimation; Hellinger integral; Radon-Nikodým densities; convergence in variation distance##### References:

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